Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose...

Chapter 6:

Series and SequencesSSMth1: Precalculus

Science and Technology, Engineering

and Mathematics (STEM)

Mr. Migo M. Mendoza

Chapter 6: Series and Sequences

Lecture 22: Series

Lecture 23: Sequences

Lecture 21: Sequences

SSMth1: Precalculus



Mr. Migo M. Mendoza

Sequences

• A sequence is a function whose domain is the set

ℤ+ ={1, 2, 3, …} of positive integers and whose range is the

set ℝ of real number.

Two Kinds of Number Sequence

1. Finite Sequence; and2. Infinite Sequence.

The Infinite Sequence

• An infinite sequence is a function whose domain is the set of positive integers. Also, it is a sequence that has no

last term.

The Finite Sequence

•A finite sequence is function whose domain consist of the first n positive integers only. In other words, it is a sequence having a

last term.

Classroom Task 1:

Direction: Identify whether the

following are finite or infinite sequence. Clap three

times if it is FINITE and once if it is INFINITE.

Example 1:

Finite or Infinite?

,...36,25,16,9,4,1

Example 2:

Finite or Infinite?

7,6,5,4,3,2,1

Example 3:

Finite or Infinite?

,...15,12,9,6,3

Example 4:

Finite or Infinite?

14,12,10,8,6,4,2

Example 5:

Finite or Infinite?

15,13,11,9,7,5,3,1

Example 6:

Finite or Infinite?

,...13,8,5,3,2,1,1

Example 7:

Write the first five terms of the sequence whose nth term is

given by the formula:

.23 nan

Final Answer:The first term of the sequence is 5, the second term is 8, the third term

is 11, and so on. The sequence can be also denoted as:

.17,14,11,8,5

Example 8:

List the four terms of the infinite sequence whose nth

term is given by:

.)3()1( 2 nc n

n

Final Answer:

The first four terms of the sequence are

.1,0,1,4

Example 9:

Find the 9th term and the 21st

term of the sequence whose nth

term is:

.12

2

n

nan

Final Answer:

The 9th term is 81/80and the 21st term is

441/440.

Performance Task 1:

Please download, print

and answer the “Let’s

Practice 21.” Kindly work

independently.

Lecture 22: SeriesSSMth1: Precalculus



Mr. Migo M. Mendoza

SeriesThe indicated sum of the terms of a

finite sequence a1, a2,a3, …, an is called series or finite series and

is denoted by:

....321 nn aaaaS

Example 10:

Find S5 where:

.423 nan

Final Answer:

S5 is 55.

Example 11:

Find S4 where :

.)1( 2na n

n

Final Answer:

S4 is 10.

Did you know?

To describe the sum of the terms of a sequence, we use the

summation notation. The Greek letter Σ (sigma) is used to indicate

each sum.

Two Kinds of Series

1. Finite Series; and2.Infinite Series

Finite Series

The sum of the first n terms of the sequence is called a finite series or the nth partial sum of

the sequence.

Finite Series

It is denoted by:

nn

n

i

i aaaaaaa

14321

1

...

Finite Series

The sum of the first n terms of the sequence is called a finite series or the nth partial sum of

the sequence.

Take Note:

i is called the index of the summation, n is the upper

limit of the summation, and 1is the lower limit of the

summation.

Something to think about…

What is the correct way of reading the summation

notation? Let’s say

n

i

ia1

Did you know?

is read as “the summation of ai where i

is from 1 to n”.

n

i

ia1

Infinite Series

The sum of all terms of the infinite sequence is called a infinite series.

Infinite Series

It is denoted by:

nn

i

i aaaaaaa

14321

1

...

Take Note:

The upper limit summation of the finite series is n,

while in infinite series is ∞(symbol for infinity).

Example 12:

Write out the terms of the following series:

5

1

22i

i

Final Answer:

11025

1

2 i

i

Example 13:

Write out the terms of the following series:

n

i

i i1

2 )3()1(

Final Answer:

)3()1(...27123)3()1( 2

1

2 ni nn

i

i

Series and Sigma Notation:

Properties of SumsSSMth1: Precalculus



Mr. Migo M. Mendoza

Property 1:

Property 1: The Sum of the First n Constant

cncn

i

1

Proof:

cncn

i

1

n

n

i

ccccccc

...54321

1

Property 2:

n

i

i

n

i

i acca11

Proof:

n

i

i

n

i

i acca11

n

n

i

i cacacacacacaca

...54321

1

]...[ 54321

1

n

n

i

i aaaaaacca

Property 3:Property 3: The Sum of the Sum or

Difference of Series

n

i

i

n

i

i

n

i

ii baba111

Proof:

n

i

i

n

i

i

n

i

ii baba111

nn

n

i

ii bababababababa

...5544332211

1

nn

n

i

ii bbbbbbaaaaaaba

...... 5432154321

1

Property 4:Property 4: The Sum of the First n

Natural Numbers

2

)1(

1

nni

n

i


Squares

6

)12)(1(

1

2

nnni

n

i


Cubes

2

1

3

2

)1(

nni

n

i

Example 14:

Use the properties of sums to evaluate the series:

4

1

73i

i

Final Answer:

2734

1

i

i

Example 15:


3

1

2 32i

ii

Final Answer:

43323

1

2 i

ii

Example 16:


100

1

223

i

i

Final Answer:

700,293,123100

1

2

i

i

Example 17:

Write the following series in summation notation:

...20161284

Final Answer:

n

i

i1

4...20161284

Performance Task 2:

Please download, print

and answer the “Let’s

Practice 22.” Kindly work

independently.

Chapter 6: Series and Sequences...The Infinite Sequence An infinite sequence is a function whose...

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